I'm wavering between taking 2nd Quantum Mechanics course and not. I've just completed the 1st course that already covered fundamentals of QC. I intend to take Quantum Information Processing along with Quantum Mechanics 2 next semester, but recently I've read some opinions that advanced QM will not be very helpful for doing (serious) research in Quantum Information/Computing. Can someone confirm whether perturbation theory has anything to do with QI/QC? I really want to expand my knowledge in QM, but the course is very challenging so I don't want to waste much time for something I'll barely use.
If you want an excellent answer, you will have to say what "2nd course in QM" means, by specifying which exact topics will be covered, other than just "perturbation theory", which is the only thing you've said about the 2nd QM course so far. Also it would be helpful for you to tell us what was covered in the first QM course.
However, I can already give a decent answer with what you've provided so far:
"Can someone confirm whether perturbation theory has anything to do with QI/QC?"
There's several benefits of learning perturbation theory.
1) Even for the "theoretical computer science" aspect of QI/QC:
- "Perturbative gadgets" first introduced by Kitaev, Kempe & Regev in 2004, were used in the proof that adiabatic quantum computers can simulate circuit-based quantum computers with polynomial overhead. They are central to proofs of QMA completeness using our wisdom of the k-local Hamiltonian problem. They are also crucial in the compiling of problems onto adiabatic quantum computers or quantum annealers.
2) At the hardware level:
If you want to understand how the hardware works, you will benefit from understanding perturbation theory, because for example, superconducting qubits (the most popular type of qubit in current large-scale hardware) are not actually qubits (2-level systems) but are oscillators with a large number of levels, and they are approximated by 2-level systems, and when choosing real physical systems with the properties (such as energy scales) required for your hardware, you could find perturbation theory very useful.
If you want to understand how to model noise/decoherence in the engineering of quantum hardware, you are likely to see perturbation theory in many possible sub-areas (ex. perturbative master equations where the system-bath coupling is considered weak).
When implementing gates, you may not physically be easily able to do the exact unitary you wish, but using perturbation theory on a known physical phenomenon could help you find a gate that closely approximates the gate you wish to implement.
3) On an academic level:
- Even if you just want to do more "hardcore" quantum information, such as studying multi-partite entanglement witnesses, quantum discord, quantum "contextuality", or thinking about interesting "quantum games" or paradoxes like the GHZ paradox, the mathematics you will learn when you study basic undergraduate level perturbation theory, will no doubt help you with doing anything involving quantum mechanics.
- You are also likely to want to be able to understand the talks at conferences, or at department seminars, or even when your colleagues/classmates in your physics/chemistry department are presenting their work, if you want to fit in well with other academics in QI/QC (academia is not just about being smart, it's about being social too).
4) Outside of QM:
- Perturbation theory is not just useful for modeling in quantum mechanics, but it's used in weather modeling, engineering, financial modeling, fluid dynamics, and even in pure mathematics (many techniques for solving or approximately solving differential equations use perturbation theory). You don't need to take a QM course to learn this, but it will give you a good appreciation for the general technique.
Aside from perturbation theory, there will be a lot of more advanced quantum mechanics and mathematical techniques you'll learn in this course that will be helpful in your career in QI/QC (or even elsewhere).
I am speaking from experience:
I personally did not take QM2 during my undergrad. I graduated from University of Waterloo and there was 6 undergrad quantum courses: QM1, QM2, QM3, Quantum Theory 1, Quantum Theory 2, and Quantum Computing. QM1, QM2, and QM3 were all mandatory for all physics degrees, and QT2 was required for others. I was also doing a degree in Biology and a degree in Mathematics and taking several arts courses in Music, Philosophy, Psychology and Anthropology, so I was very protective about wasting my time. I felt that QM1 and Quantum Computing was enough, so I skipped QM2, QM3, QT1, QT2, and later found out that I missed a lot:
- Variational Principle (which is now extremely popular if you are to look at the VQE algorithm or the variational model for quantum computing).
- Advanced mathematics of spin (Clebsch-Gordon coefficients, adding angular momenta, etc.).
- Perturbation theory (as you mentioned yourself, and I went into detail describing above).
- So much that I don't even know that I missed. I wasn't there, so I don't even know what I don't know. This has caused me to lose a lot of confidence when working in quantum computing, because I wasn't aware whether or not I was at some disadvantage due to everyone else having learned something about a topic on which I was working.
"I really want to expand my knowledge in QM, but the course is very challenging so I don't want to waste much time for something I'll barely use."
You said you want to do "reasearch" in quantum computing. Most research is done at the post-PhD level, and much of the rest of it is done as a post-graduate student (Masters or PhD student). You can do some research as an undergraduate student too, but for most of your life you will not be a student (whether it's undergrad, masters, or PhD) so really, except in very rare cases, most of your career will be at the post-PhD level if you want to be a researcher. As an undergrad, it's not easy to see what things will be like 10 years later as a researcher, so you came to the right place for advice. My advice is to keep a more open mind about what might be a "waste of time", "barely useful", or "too challenging". At any university, QM2 is not considered extremely advanced if you want to do research in quantum information (QM1, QM2, and QM3 are all mandatory for anyone to get a physics degree at University of Waterloo, for example).
In one of the comments, someone said:
"spherical harmonics/calculating hydrogen orbitals/perturbation theory are not useful for quantum computing."
- I have addressed perturbation theory already.
- Spherical harmonics and solving the Schroedinger equation for the hydrogen atom are certainly useful as QCQC (quantum computing for quantum chemistry) has now become one of the most popular areas in the field, since it's probably the area where QC might be able to have the most real-world impact (factoring numbers, or Shor's algorithm, is not actually useful, since people are already switching to post-quantum cryptography). Paragraph 3 of this answer and the citation marked as "reference 1" claim that QCQC is the one area where quantum computing is "not just hype".
- If you want to choose research as a career, also try not to put 100% of your eggs in the QI/QC basket, because a "quantum winter" may come, where popularity of quantum computing decays. If I had to guess right now, I'd say the next big thing will be energy science as our world starts to become more worried about running out of resources and reducing our ecological footprint: you may wish to study condensed matter physics so that you remain employable in the materials science, materials design, energy storage, energy transport, and energy conversion fields (where people design long lasting batteries, better solar cells, more energy efficient smartphone screens, stronger materials for cars, bikes, planes, spaceships, etc.). If you want to remain employable in any of these fields, or to at least be able to understand what's going on when the next huge discoveries are published, you will need an understanding of QM2, QM3 and more.